Three-dimensional X-ray imaging is based on taking several 1-D or 2-D projection images of a 3-D body from different directions. If 1-D projection images are available from all around a 2-D slice of the body with dense angular sampling, the inner structure of the slice can be determined. This is known as Computerized Tomography (CT) imaging technology, which is widely used in medicine today. A crucial part of CT technology is the reconstruction algorithm taking the X-ray images as argument and producing a voxel representation of the 3-D body.
In many practical cases X-ray projection images are available only from a limited angle of view. A collection of X-ray images of a 3-D body is called sparse projection data if (a) the images are taken from a limited angle of view or (b) there are only a small number of images. Sparse projection data does not contain sufficient information to completely describe the 3-D body.
However, some a priori information about the body is typically available without X-ray imaging. Combining this information with sparse projection data enables more reliable 3-D reconstruction than is possible by using only the projection data.
Traditional reconstruction algorithms such as filtered backprojection (FBP), Fourier reconstruction (FR) or algebraic reconstruction technique (ART) do not give satisfactory reconstructions from sparse projection data. Reasons for this include requirement for dense full-angle sampling of data, difficulty to use a priori information, for example nonnegativity of the X-ray attenuation coefficient, and poor robustness against measurement noise. For example the FBP method relies on summing up noise elements with fine sampling, leading to unnecessarily high radiation dose.
The traditional setting of the image reconstruction problem in tomography assumes that the imaging geometry is known accurately. The prior art requires either reference feature points, with which the unknown part of the imaging geometry (e.g. motion of the imaging device or the object) is computed, or precise prior knowledge of the imaging geometry. Anyway the prior art does not know effective methods to determine the imaging geometry like the motion of the imaging device and/or the object, though all parts of the imaging geometry are rarely precisely known in advance.